Extending ground-motion prediction equations for spectral ...

Bull Earthquake Eng (2012) 10:379–399 DOI 10.1007/s10518-011-9304-0 ORIGINAL RESEARCH PAPER

Extending ground-motion prediction equations for spectral accelerations to higher response frequencies Julian J. Bommer · Sinan Akkar · Stéphane Drouet

Received: 23 April 2011 / Accepted: 5 July 2011 / Published online: 16 July 2011 © Springer Science+Business Media B.V. 2011

Abstract Ground-motion prediction equations (GMPEs) for spectral accelerations have traditionally focused on the range of response periods most closely associated with the dynamic characteristics of buildings. Providing predictions only in this period range (from 0.1 to 2 or 3 s) has also accommodated the assumed limitations on the usable period range resulting from the processing of accelerograms. There are, however, engineering applications for which estimates of spectral ordinates are required at shorter response periods. Recent work has demonstrated that high-frequency spectral ordinates are relatively insensitive to record processing, contrary to previous assumptions. In the light of this finding, additional regressions are performed to extend a recent pan-European GMPE to higher response frequencies. This model and others that also include coefficients for spectral ordinates at several high response frequencies are used to explore options for interpolating coefficients for equations that do not provide good coverage in this range. The challenges and uncertainties associated with such interpolations are discussed. The paper concludes that a set of standard response frequencies could be usefully established for future GMPEs. Keywords Ground-motion prediction equations · Spectral acceleration · Short response periods · High-frequency ground motion 1 Introduction Ground-motion prediction equations (GMPEs) for the estimation of response spectral ordinates at several response periods have become a fundamental requirement for probabilistic J. J. Bommer (B) Civil & Environmental Engineering, Imperial College, London, SW7 2AZ, UK e-mail: [email protected] S. Akkar Earthquake Engineering Research Centre, Middle East Technical University, Ankara 06800, Turkey S. Drouet Geoter International, 13360 Roquevaire, France

123

380

Bull Earthquake Eng (2012) 10:379–399

seismic hazard analysis (PSHA) ever since it was first recognized that the practice of anchoring fixed spectral shapes to site-specific values of peak ground acceleration (PGA) is unlikely to yield consistent hazard levels at all response periods (McGuire 1977). The range of response periods covered by empirical GMPEs has been determined as a balance between engineering requirements and the limitations imposed as a consequence of the filters applied in processing the strong-motion accelerograms. Considerable attention has been given in recent years to extending the upper limit of the period range covered by predictive equations for response spectral ordinates, in order to address the input needed for direct displacement-based approaches as well as for the analysis of high-rise buildings, long-span bridges and base-isolated structures. The first pan-European GMPEs for spectral accelerations by Ambraseys et al. (1996) only provided coefficients for periods up to 2.0 s. The model of Ambraseys et al. (2005) extended this to 2.5 s. Berge-Thierry et al. (2003) presented European GMPEs for predictions up to the 10-s response period, but since the accelerograms had been filtered with a cut-off at 4 s the equations for periods beyond 3 s are unusable (Boore and Bommer 2005). Akkar and Bommer (2006) individually processed records from the European Strong-Motion Database (Ambraseys et al. 2004) and established criteria for defining the usable period range. Akkar and Bommer (2007b) used these records to derive predictive equations for spectral displacements up to 4 s, although in revising this model Akkar and Bommer (2010) focused on pseudo-spectral acceleration and chose 3 s as a more appropriate upper limit. In contrast, the GMPEs developed in the Next Generation Attenuation (NGA) project provide predictions of response spectral ordinates up to 10 s, although it is noted that from PGA to this period there is a more than five-fold reduction in the number of usable records (Abrahamson and Silva 2008). In the short-period range, a parallel story has been unfolding over the last 15 years. Tables 1, 2 and 3 summarise the high-frequency characteristics of GMPEs for active crustal, subduction and stable continental regions, respectively. The tables include models in current usage except that in Table 1 a number of European models that have been superseded are also included to provide a more complete overview for this region. From the tables it can be appreciated that the minimum response period considered has varied considerably, as has the resolution of the short-period ordinates. With the exception of the hybrid stochastic-empirical model of Douglas et al. (2006) for southern Spain, the European GMPEs do not provide predictions for spectral accelerations at periods less than 0.03 s, and for the pan-European models (as opposed to those applicable to a single country) the limit is 0.05 s. In Ambraseys et al. (1996)—the first pan-European GMPE for response spectral ordinates—the authors opted for a lower limit of 0.1 s on the basis of not being able to apply instrument corrections to a large proportion of their dataset because the transducer characteristics were unknown. They also reasoned that this limitation was acceptable since for structural engineering applications, shorter response periods were not particularly important. Neither of these premises holds currently. As discussed in the next section, the short-period spectral ordinates are much less sensitive to strong-motion record processing than previously assumed, which allows reliable response ordinates to be calculated at short periods. Regarding the engineering needs for short-period spectral accelerations, there are at least two particular applications for which these become important. The first corresponds to the seismic analysis and design of rigid structures, systems and components, particularly in nuclear power plants; the issue of correctly estimating high-frequency ground motions for such facilities has been highlighted in recent years (e.g., USNRC 2008). The second issue is the definition of the vertical response spectrum, which generally peaks at much shorter periods than the horizontal motion (e.g., Bozorgnia and Campbell 2004). This feature is particularly important in dynamic structural analysis since the natural vibration periods of structures in the verti-

123


381

Table 1 Short-period characteristics of GMPEs for spectral accelerations in active regions of shallow crustal seismicity Model

Region

PGA Tmin N1 (T < 0.1 s)

N2 (0.1 ≤ T < 0.2 s)

Sampling

Sabetta and Pugliese (1996)

Italy

Y

0.04

2

2

Log(f) Uneven

Ambraseys et al. (1996)

Europe and the Middle East Europe, Middle East and California Turkey

Y

0.10

0

10

Linear Uneven

N

0.03 28

24

Log(f) Uneven

Berge-Thierry et al. (2003)

Kalkan and Gülkan (2004)

Y

0.10

0

10

Log(T) Uneven

Y

0.10

0

2

Log(T) Uneven

Y

0.10

0

10

Log(T) Uneven

Y

0.05 10

10

Linear Uneven

Douglas et al. (2006)

Northwestern Turkey Extensional regions Europe and the Middle East Southern Spain

N

0.01

5

2

Log(T) Uneven

McVerry et al. (2006)

New Zealand

Y

0.075 1

1

Log(T) Uneven

Kanno et al. (2006)

0.05

6

Log(T) Uneven

Zhao et al. (2006)

Japan (plus some Y additional data) Japan Y

0.05

1

2

Log(T) Uneven

Danciu and Tselentis (2007)

Greece

Y

0.10

0

2

Log(T) Uneven

Cotton et al. (2008)

Japan

Özbey et al. (2004) Pankow and Pechmann (2004) Ambraseys et al. (2005)

5

N

0.01

7

3

Log(T) Uneven

California, Taiwan, Europe, Middle East Boore and Atkinson (2008) California, Taiwan, Europe, Middle East Campbell and Bozorgnia (2008) California, Taiwan, Europe, Middle East Chiou and Youngs (2008) California, Taiwan, Europe, Middle East Idriss (2008) California, Taiwan, Europe, Middle East Cauzzi and Faccioli (2008) Japan, Italy, California, Iceland, Iran Massa et al. (2008) Northern Italy

Y

0.01

6

2

Log(T) Uneven

Y

0.01

5

2

Log(T) Uneven

Y

0.01

5

2

Log(T) Uneven

Y

0.01

5

2

Log(T) Uneven

Y

0.02

6

2

Log(T) Uneven

Y

0.05

1

2

Linear Even

Y

0.04

2

2

Log(T) Uneven

Bindi et al. (2009)

Italy

Y

0.03

3

2

Log(T) Uneven

Akkar and Bommer (2010)

Europe and the Middle East Turkey

Y

0.05

1

2

Linear Even

Y

0.03

3

2

Log(T) Uneven

Abrahamson and Silva (2008)

Akkar and Ça˘gnan (2010)

PGA, Y if coefficients explicitly provided for peak ground acceleration; Tmin , minimum response period covered, excluding PGA; N1 , the number of response periods (excluding PGA) less than 0.1 s for which coefficients are provided; N2 , the number of response periods in the 0.1–0.2 s interval for which coefficients are provided; Sampling of response periods covered, and whether based on even or uneven (over entire range covered) intervals of period, or logarithm of the period (T) or frequency (f)

123

382


Table 2 Short-period characteristics of GMPEs for spectral accelerations in subduction regions PGA Tmin

Region

Youngs et al. (1997)

Worldwide

Y

0.075 1

1

Log(T) Uneven

Atkinson and Boore (2003)

Worldwide

Y

0.04

1

1

Log(f)

Uneven

García et al. (2005)

Central Mexico

Y

0.04

3

1

Log(f)

Uneven

Kanno et al. (2006)

Y

0.05

5

6

Log(T) Uneven

Zhao et al. (2006)

Japan (plus some additional data) Japan

Y

0.05

1

2

Log(T) Uneven

McVerry et al. (2006)

New Zealand

Y

0.075 1

1

Log(T) Uneven

0.01

7

4

Log(T) Uneven

0.05

3

3

Log(f)

Lin and Lee (2008)

Northern Taiwan Y (plus other data) Atkinson and Macias (2009) Cascadia Y

N1 (T < 0.1 s)

N2 (0.1 ≤ T < 0.2 s)

Model

Sampling

Even

See Table 1

Table 3 Short-period characteristics of GMPEs for spectral accelerations in Stable Continental Regions Region

Toro et al. (1997)

Eastern North America Y*

0.03 2

1

Campbell (2003)

Eastern North America N

0.01 5

2

Log(T) Uneven

Tavakoli and Pezeshk (2005) Eastern North America Y*

0.05 2

2

Log(T) Uneven

Douglas et al. (2006)

0.01 5

2

Log(T) Uneven

Atkinson and Boore (2006) Eastern North America Y*

0.025 6

4

Log(f) Even

Atkinson (2008)

0.10 0

1

Log(T) Uneven

Southern Norway

PGA Tmin N1 (T < 0.1 s)

N2 (0.1 ≤ Sampling T < 0.2 s)

Model

N

Eastern North America Y

Log(f) Uneven

See Table 1 * For these models, the paper does not make it clear whether the PGA values have been taken directly from recordings or inferred from the spectral acceleration at a very short response period (e.g., 0.01 s)

cal direction may be much smaller than in the horizontal direction (Elnashai and Papazoglu 1997). Within the framework of PSHA, it is now generally regarded more appropriate to obtain vertical response spectra by applying vertical-to-horizontal (V/H) ratios to the horizontal spectra rather than conducting hazard calculations directly in terms of the vertical accelerations (e.g., Gülerce and Abrahamson 2011; Bommer et al. 2011a). Therefore, the horizontal response spectrum must be specified with a suitable degree of resolution at short response periods. As noted above, in the next section of the paper, recent insights regarding the usable range of spectral ordinates at short periods is taken advantage of to extend the Akkar and Bommer (2010) model to periods below 0.05 s. The remainder of the paper then explores the interpolation of missing coefficients for when GMPEs with insufficient resolution at short periods are used in the kind of applications mentioned above. The first issue addressed is the critical question of deciding at what response period, PGA and the spectral acceleration become equivalent. Options and constraints on interpolating between this value and the shortest period provided are then examined in the penultimate section. The paper closes with a brief discussion and succinct conclusions.

123


383

At this point it is worth making a note regarding nomenclature: although in structural engineering practice it is more common to refer to response periods, we also refer to the reciprocal quantity of response frequency, which is more commonly used in seismology as well as in nuclear engineering applications. Since the European equations of Akkar and Bommer (2010) extended in the following section are expressed in terms of response period, the various graphs are plotted in terms of this parameter, with a second x-axis above each plot with the equivalent response frequencies.

2 High-frequency extension of a European ground-motion model Douglas and Boore (2011) have recently shown that high-frequency spectral ordinates are much less sensitive to high-cut filters than the low-frequency accelerations. This prompted a re-examination of the limit of 20 Hz (0.05 s) chosen by Akkar and Bommer (2010) for the derivation of GMPEs for pseudo-spectral accelerations in Europe and the Middle East. The strong-motion dataset used for this study is exactly the same as that employed by Akkar and Bommer (2010). The records are from Europe, the Middle East and the Mediterranean, and have been obtained from the European strong-motion database (Ambraseys et al. 2004). The characteristics of the dataset in terms of magnitude, style-of-faulting and number of records from each event are listed in Akkar and Bommer (2007a). Whereas Akkar and Bommer (2007b, 2010) only applied low-cut filters to the records, for this study each record was also subjected to an individually selected high-cut filter, where necessary. The criteria for selecting these high-cut filters are discussed in detail by Akkar et al. (2011), who also discuss the criterion established for selecting the lower limit for usable periods of the band-pass filtered records. Whereas for regressions on spectral ordinates at long periods, Akkar and Bommer (2010) removed records individually at periods beyond their usable limit, for the short-period equations it was decided to remove any record that failed the criterion at 0.05, 0.04, 0.03, 0.02 or 0.01 s from the regressions at each of these periods, as well as for PGA (Akkar et al. 2011). For periods of 0.10 s and greater, the high-cut filter was found to exert no influence at all, so the coefficients presented by Akkar and Bommer (2010) at those periods remain unchanged. For the short-period spectral predictions, a total of 49 records were removed from the database of 532 accelerograms, which is comparable to the number of records used for the Akkar and Bommer (2010) at response periods on the order of 2.5 s. The distributions of both the 49 discarded records and the remaining 483 records in terms of magnitude, distance and site class are shown in Fig. 1. From these plots it can be appreciated that the distribution of the records available for the regressions on short-period spectral ordinates is not significantly altered by the removal of the small number of records rejected (less than 10% of the total database), which show no particular clustering in terms of magnitude, distance or site class, and interestingly are also not exclusively from analog instruments. An important point to note is that instrument transducer corrections were not applied to the records, for a variety of reasons. The recordings from digital accelerographs are considered to not require transducer corrections, and for many of the analog instruments the actual transducer characteristics (frequency and damping) are not known. We are also cautious about applying an instrument correction in any case since whichever method is used these corrections essentially amplify high-frequency components of the motion, which will affect high-frequency noise as well as the high-frequency components of the motion diminished by the transducer response (Boore and Bommer 2005). Although for analog instruments such as the SMA-1 accelerograph, Fourier amplitudes at frequencies above about 25 Hz may be

123

384


Fig. 1 Distribution in terms of magnitude, distance and site classification of the records retained for regressions at short periods (left) and those rejected because of the strong influence of the high-cut filter (right) Table 4 Definition of site class dummy variables in Eq. (1)

Site class

Vs30 range

SS

SA

Rock Stiff soil

>750 m/s

0

0

360–750 m/s

0

Soft soil

1

7 are still relatively sparse. Moreover, all of the records from events of Mw 7.2 and above have been generated by strike-slip events (primarily the 1990 Manjil earthquake in Iran and the 1999 Kocaeli and Düzce earthquakes in Turkey) with the exception of the 1978 Tabas earthquake in Iran, which was associated with a low-angle reverse rupture. There are only three records in the Akkar and Bommer (2010) database from this Mw 7.3 thrust earthquake, and two of these are excluded from the regressions at short period (Fig. 1).

3 Peak ground acceleration and high-frequency spectral accelerations If the highest frequency for which a GMPE includes coefficients is smaller than that needed for a particular application, the possibility exists to interpolate the missing coefficients if coefficients are provided for PGA and if these can be associated with a particular response frequency. As can be appreciated from Table 1, this is a potentially useful option since only three models lack coefficients for PGA and two of these provide predictions of spectral ordinates at 100 Hz. The challenge lies in deciding at what response period the spectral acceleration becomes equal to PGA. The plots in Fig. 4 seem to indicate that for most magnitudes and distances, the spectrum does not become flat at short periods, with the ordinates at 50 Hz (0.02 s) being consistently higher than those at 100 Hz (0.01 s). Figure 5 shows the actual data in terms of the correlations between PGA and PSA at the four shortest response periods considered, from which it can be clearly seen that the two become essentially equivalent at 100 Hz. Since sampling rates for accelerograms rarely exceed 200 intervals per second, the Nyquist frequency is generally limited to 100 Hz, and it is possible that these figures might change if the records had higher sampling rates. However, any such effect is likely to be small since the high-frequency

123

388


Fig. 4 Predicted median spectral ordinates (black lines) from the extended model for various combinations of the explanatory variables. The dark symbols are predicted PGA values; the open symbols are predictions obtained from the coefficients of Akkar and Bommer (2010) for PGA and 0.05 s

spectral response ordinates are not strongly influenced by the high-frequency components of the motion (Douglas and Boore 2011; Akkar et al. 2011). Even at 100 Hz, however, the match is not exact although for two-thirds of the records the difference is less than 1%, and with the exception of just two records the differences are always smaller than 5%. Idriss (2007) found similar results for the NGA dataset, with 97.5% of the records showing differences less than 2% between the two parameters; for the other records, all from California, Idriss (2007) reported differences of between 3 and 4%. Visual inspection of the handful of records for which the differences between PGA and PSA(0.01s) are not negligible fails to reveal any reasons to discard the records from

123


389

Fig. 5 Comparison of PGA and PSA values at short periods for the 483 records used in the regressions to obtain the coefficients in Table 5. The straight lines indicate equivalence rather than a fit to the data; the r values are the correlation coefficients between each pair of data points

regressions. It is also interesting to note that the few records for which PSA at 100 Hz is slightly greater than PGA are not exclusively from rock sites, although this observation needs to be tempered by the fact that the site classifications of many strong-motion recording stations in Europe and the Middle East can be rather unreliable (e.g., Ambraseys et al. 2004). In Table 5, T = 0.00 s indicates the coefficients for PGA, for which we performed separate regressions using the same records retained for the regressions at response periods between 0.01 and 0.05 s. In describing the NGA database, Chiou et al. (2008) state that for western North America, PGA and the 5%-damped spectral acceleration at 0.01 s are equivalent, even for hard rock locations. This is supported by the fact that all but one of the NGA models report identical coefficients for the equations predicting PGA and PSA(0.01s). Only Boore and Atkinson (2008) report very slightly different coefficients for these two quantities, which is a result of constraining the pseudo-depth term, h, for response periods from 0.01 to 0.05 s to the value obtained by regression for PGA.

123

390


Fig. 6 Comparison of predicted values of PGA and pseudo-spectral acceleration (PSA) at 0.01 s using Eqs. (1) and (2) and the coefficients in Table 5. The curves are plotted using a linear vertical axis in order not to conceal any differences between the pairs of curves

In Table 5, it can be seen that the coefficients obtained independently for PGA and for PSA(0.01s) are very slightly different. In order to investigate whether or not these minor differences in the coefficients actually correspond to differences in the predicted quantities, Figure 6 compares predicted values of PGA and PSA(0.01s) for various combinations of the predictor variables, including the limiting values of magnitude, and also for different exceedance probabilities (medians and 84-percentiles). The 0.01 s pseudo-spectral accelerations are very marginally greater than the PGA values, but always by less than 1.5% and even then only at short distances (for Mw 5 beyond 20 km and for Mw 7.6 at all distances the difference is less than 1%). From this we conclude that the two quantities can indeed be treated as equivalent, and in most of the remaining figures in this paper we present only the spectral acceleration at 0.01 s and not PGA. Standard elastic design spectra used in the nuclear industry are generally flat at higher frequencies, implying an assumed equivalence between pseudo-spectral accelerations at higher frequencies and PGA (e.g., Bommer et al. 2011b). In these standard nuclear industry spectra, spectral acceleration generally becomes equal to PGA at all response frequencies above 33 Hz, which seems rather low in light of the observations discussed above. This may explain the contentions that have arisen with regards to such standard design spectra for nuclear facilities (USNRC 2008), as discussed in the Introduction. Insights into the response frequency at which PSA becomes equivalent to PGA can be obtained from GMPEs that provide a PGA model together with equations for response spectral ordinates at several high frequencies. Plots illustrating such an approach are not included for reasons of space, but the NGA models constitute a useful resource for such an exercise. These models show that the site classification influences the high-frequency shape of the spectrum, with the spectra becoming effectively flat above 50 Hz for soft sites. For stiff and hard sites, however, the 50 Hz ordinate remains higher than that at 100 Hz, albeit very slightly so. As Table 3 indicates, the available Eastern North American equations are not particularly amenable to such an exercise because they either lack a PGA equation or are limited to

123


391

rather small maximum frequencies. However, plots such as Figure 3a of Campbell (2003) and Figure 1 of Douglas et al. (2006) strongly suggest that for the high stress-drop, high shearwave velocity (2,000 m/s and greater) and low kappa conditions of Eastern North America, their 50 Hz ordinate is much higher than the 100 Hz pseudo-spectral acceleration, and it is not entirely clear if even at the latter value the PSA and PGA values are equivalent. These observations suggest that for interpolating coefficients for high-frequency spectral ordinates, PGA should be assumed to correspond to the spectral acceleration at 100 Hz, since adopting a smaller response frequency may be non-conservative. However, for very hard rock conditions typical of Stable Continental Regions (SCR), it may be the case that even 100 Hz is not sufficiently high for anchoring PGA as the reference point for interpolations.

4 Options and issues for interpolation at high response frequencies For any application in which the high-frequency spectral accelerations are of importance, the lack of high-frequency resolution in many current GMPEs (Tables 1, 2, 3) may necessitate interpolation. This is particularly likely to be the case in the context of PSHA, where the imperative of capturing epistemic uncertainty leads to several GMPEs being combined in a logic-tree framework (Bommer et al. 2005), which will obviate the option of choosing only GMPEs with coefficients at the required target frequencies. Once faced with such a situation, there are two options available to the hazard analyst, namely to interpolate for the missing spectral ordinates or to interpolate for the missing coefficients. Although the former may be more convenient, it offers no flexibility and would need to be repeated for every uniform hazard, scenario or conditional mean spectrum of interest, and it would not allow disaggregation at the missing frequencies. For these reasons, we focus instead on the more general solution of interpolating the coefficients of the equations. For some recent models with complex functional forms and large numbers of coefficients, this could become a little cumbersome although it is also the case that such models are generally not those lacking high-frequency resolution. In order to interpolate (as opposed to extrapolate, which is strongly discouraged in both the high- and low-frequency ranges), the first step is to fix the end point. This will usually involve taking the coefficients for the prediction of PGA as a surrogate for the spectral ordinate predictions at 100 Hz, as discussed in the previous section. The next step is to decide whether the interpolation will be against the natural values of frequency (or period) or their logarithms. Using the GMPEs of Akkar and Bommer (2010) and its high-frequency extension summarised in Table 5, Toro et al. (1997) and Campbell (2003) as examples, a clear and simple conclusion is reached: if the original coefficients are for frequencies defined in log-space, then the coefficients should be interpolated against log(f) or log(T), whereas for those giving coefficients at frequencies in linear space, the interpolation should be directly against frequency or period. Referring to the penultimate columns in Tables 1, 2 and 3, the indication is that for all but a handful of models, the interpolation will be performed against the logarithm of frequency or period. Figure 7 shows the results of not following this simple rule and the results of interpolating coefficients for Akkar and Bommer (2010)—as if the high-frequency extension presented in this paper had not been performed—against log(T). Equally poor results could be displayed for interpolations of the coefficients from either of the ENA models against period rather than its logarithm. Figure 8 shows the same information as Fig. 7, but in this case a log-linear interpolation has been used instead of cubic spline. The results for the wide interpolation gap (0.1 to 0.01 s) obtained with the log-linear interpolation are clearly superior to those obtained with

123

392


Fig. 7 Median spectral ordinates for rock sites at 20 km from earthquakes of different magnitudes obtained directly from the equations of Akkar and Bommer (2010) and the high-frequency extensions in Table 5, compared with ordinates from interpolated coefficients using a cubic spline against log(T)

the cubic spline. Figure 9 shows the results obtained interpolating the coefficients against linear period, and as in Fig. 7 using various intervals between 100 Hz (PGA) and the highest frequency assumed to be covered by the equation. Of course, for this model the interpolations are performed as if the high-frequency coefficients in Table 5 had not been derived, in order to provide an independent check. The results in Fig. 9 are uniformly better than those in Figs. 7 and 8, including for the case of the widest interval of missing coefficients. The conclusion from this would seem to be that the interpolation should always be made against period or its logarithm, depending on the spacing of the original equations. In both Figs. 7 and 9, and others that follow, the interpolations are made using a cubic spline, which is a piecewise polynomial function. There are many options, including various polynomials, but the cubic spline is found to be particularly well suited to this purpose. The results in Fig. 9 suggest even if the highest frequency covered by the coefficients of the Akkar and Bommer (2010) model had been 10 Hz, satisfactory results could be obtained by interpolation. There is no marked improvement in the results when it is assumed that the highest frequency for which coefficients are included is 25 or 20 Hz. The likely explanation for this observation is that the peak in the response spectrum for this model is always at a frequency lower than 10 Hz, which means that in all these cases the interpolation is being made in the descending branch of the acceleration spectrum to where it converges with PGA. A different situation, however, is encountered with the Eastern North American GMPEs of Toro et al. (1997) and Campbell (2003) for which similar interpolation plots are shown in Figs. 10 and 11 respectively. In these cases, it is clear that if the highest frequency for which coefficients were included were only 10 Hz, then interpolation between this value and 100 Hz (assuming that this is still an appropriate anchor of PGA even for such hard rock sites)

123


393

Fig. 8 As for Fig. 7 but using log-linear interpolation

Fig. 9 Median spectral ordinates for rock sites at 20 km from earthquakes of different magnitudes obtained directly from the equations of Akkar and Bommer (2010) and the high-frequency extensions in Table 5, compared with ordinates from interpolated coefficients using a cubic spline against linear period

123

394


Fig. 10 Median spectral ordinates for hard rock sites at 20 km from earthquakes of different magnitudes obtained directly from the equations of Toro et al. (1997), compared with ordinates from interpolated coefficients using a cubic spline against logarithm of period applied over different ranges

would produce very erroneous results. The same rationale given above still holds: the peak of the spectra in these cases is at frequencies greater than 10 Hz. Whereas from the Toro et al. (1997) results in Fig. 10 it appears that 25 Hz is an adequate starting point, the results for Campbell (2003) in Fig. 11 suggest that interpolation beyond 33 Hz is acceptable (starting at 20 Hz also produces erroneous results), Figure 12 shows the results obtained with the Campbell (2003) equations if log-linear interpolation rather than a cubic spline is employed to estimate the missing coefficients, from which it is immediately apparent that results are very poor. This supports the recommendation to always use a cubic spline and interpolation against the periods spaced either linearly or logarithmically according to the response periods (or frequencies) at which the existing coefficients are provided.

5 Discussion and conclusions The rule-of-thumb that the natural period of reinforced concrete buildings can be estimated from the number of storeys divided by ten has led to the common perception that only spectral accelerations at frequencies of 10 Hz or less are of interest to structural engineering. However, there are many applications, particularly in the seismic design and assessment of nuclear power plants, for which spectral accelerations at higher response frequencies become important. This is all the more so when vertical spectral accelerations are required. Since many GMPEs do not provide predictions of ordinates at high frequencies, their employment for such applications necessitates interpolation of the missing coefficients. This

123


395

Fig. 11 As for Fig. 10 but using the equations of Campbell (2003)

Fig. 12 As for Fig. 11 but using log-linear interpolation

123

396


is possible if in addition to coefficients for spectral ordinates at various response frequencies the model includes coefficients for the prediction of PGA. The assumption should be made that PGA is equivalent to the spectral acceleration at 100 Hz, although it should be noted that for very hard rock conditions typical of Eastern North America this may be non-conservative. Interpolation should then be performed using a cubic spline against frequency (or period) or its logarithm in accordance with the spacing of the response frequencies at which coefficients are provided. This can be expected to produce reasonable results for GMPEs from active regions even if the highest response frequency modelled is just 10 Hz, but for very hard rock SCR models interpolation will not yield reliable results unless the equation includes coefficients for the spectral ordinates at 25 or preferably 33 Hz at least. If an SCR GMPE only provides coefficients at lower frequencies, there is little chance of obtaining estimates of high-frequency spectral accelerations. The key issue is whether or not coefficients are provided for a response frequency above that at which the peak of the response spectrum occurs; if not, then no interpolation scheme can provide usable estimates of the missing coefficients. The widespread belief that reliable response spectral ordinates cannot be obtained from strong-motion recordings from older accelerogaphs (or that have not had a transducer correction provided) or have been high-cut filtered, has recently been shown to be incorrect. The reason is very simply that the high-frequency response spectral ordinates are generated by lower-frequency components of the record; in other words, the response spectral ordinates at high frequencies are not strongly influenced by the high-frequency components of the Fourier amplitude spectrum. Therefore, all developers of ground-motion prediction equations should be encouraged to include coefficients for spectral ordinates for frequencies up to 50 Hz, if possible; this could be added to the list of desirable features of GMPEs listed by Bommer et al. (2010). Additional benefit could be afforded by the establishment of a standard series of reference response frequencies or periods, probably sampled in log-space, as an update to the linearly spaced periods (with the spacing widening across successive intervals of longer periods) used for the CalTech Blue Book series (Brady et al. 1973) that covered the interval from 0.04 to 15.0 s, or a frequency range from 0.067 to 25 Hz. Although most researchers in this field would argue that the upper limit of 15 s was very optimistic, the lower limit could easily be extended at least down to 0.02 s, which in conjunction with PGA (that can be assumed equal to PSA at 0.01 s) would provide adequate coverage of high-frequency spectral ordinates. Such a scheme of standardisation should probably also include a decision to use either pseudo-absolute or absolute acceleration spectra, since the two are not identical. Most of the GMPEs listed in Tables 1, 2, 3 predict pseudo-spectral ordinates, which offer the advantage of compatible acceleration (at high frequencies) and displacements (at low frequencies). A key consideration in the definition of any sampling scheme for target response frequencies is that it should be sufficiently dense in the higher frequency range to ensure that the peak in the response spectrum is always clearly and accurately defined in the predicted spectral shapes. Acknowledgments The work presented in this paper has been mainly developed within the SHARE (Seismic Hazard Harmonization in Europe) Project funded under contract 226967 of the EC-Research Framework Programme FP7. The paper was greatly improved through constructive and insightful reviews by Drs. Laurentiu Danciu and John Douglas. The authors are grateful to Dr. Peter Stafford for drawing the cubic spline to our attention as a promising interpolation function. We are also grateful to Prof. Frank Scherbaum for useful suggestions and for sharing his evolving insights into the relationship between response spectra and Fourier amplitude spectra.

123


397

References Abrahamson NA, Silva WJ (2008) Summary of the Abrahamson & Silva NGA ground-motion relations. Earthq Spectra 24(1):67–97 Akkar S, Bommer JJ (2006) Influence of long-period filter cut-off on elastic spectral displacements. Earthq Eng Struct Dyn 35(9):1145–1165 Akkar S, Bommer JJ (2007a) Empirical prediction equations for peak ground velocity derived from strongmotions records from Europe and the Middle East. Bull Seismol Soc Am 97(2):511–530 Akkar S, Bommer JJ (2007b) Prediction of elastic displacement response spectra at multiple damping levels in Europe and the Middle East. Earthq Eng Struct Dyn 36(10):1275–1301 Akkar S, Bommer JJ (2010) Empirical equations for the prediction of PGA, PGV and spectral accelerations in Europe, the Mediterranean and the Middle East. Seismol Res Lett 81(2):195–206 Akkar S, Ça˘gnan Z (2010) A local ground-motion predictive model for Turkey and its comparison with other regional and global ground-motion models. Bull Seismol Soc Am 100(6):2978–2995 Akkar S, Kale Ö, Yenier E, Bommer JJ (2011) The high-frequency limit of usable response spectral ordinates from filtered analogue and digital strong-motion accelerograms. Earthq Eng Struct Dyn. doi:10.1002/ eqe.1095 (in press) Al Atik L, Abrahamson NA, Bommer JJ, Scherbaum F, Cotton F, Kuehn N (2010) The variability of groundmotion prediction models and its components. Seismol Res Lett 81(5):783–793 Ambraseys NN, Simpson KA, Bommer JJ (1996) The prediction of horizontal response spectra in Europe. Earthq Eng Struct Dyn 25:371–400 Ambraseys NN, Smit P, Douglas J, Margaris B, Sigbjörnsson R, Ólafsson S, Suhadolc P, Costa G (2004) Internet site for European strong-motion data. Bollettino di Geofisica Teorica ed Applicata 45(3):113–129 Ambraseys NN, Douglas J, Smit P, Sarma SK (2005) Equations for the estimation of strong ground motions from shallow crustal earthquakes using data from Europe and the Middle East: Horizontal peak ground acceleration and spectral acceleration. Bull Earthquake Eng 3(1):1–35 Atkinson GM (2008) Ground-motion prediction equations for Eastern North America from a referenced empirical approach: implications for epistemic uncertainty. Bull Seismol Soc Am 98(3):1304–1318 Atkinson GM, Boore DM (2003) Empirical ground-motion relations for subduction-zone earthquakes and their application to Cascadia and other regions. Bull Seismol Soc Am 93(4):1703–1729 Atkinson GM, Boore DM (2006) Earthquake ground-motion prediction equations for Eastern North America. Bull Seismol Soc Am 96(6):2181–2205 Atkinson GM, Macias M (2009) Predicted ground motions for great interface earthquakes in the Cascadia subduction zone. Bull Seismol Soc Am 99(3):1552–1578 Berge-Thierry C, Cotton F, Scotti O, Griot-Pommera D-A, Fukushima Y (2003) New empirical spectral attenuation laws for moderate European earthquakes. J Earthq Eng 7(2):193–222 Bindi D, Luzi L, Pacor F, Sabetta F, Massa M (2009) Towards a new reference ground motion prediction equation for Italy: update of the Sabetta-Pugliese (1996). Bull Earthquake Eng 7(3):591–608 Bommer JJ, Scherbaum F, Bungum H, Cotton F, Sabetta F, Abrahamson NA (2005) On the use of logic trees for ground-motion prediction equations in seismic hazard assessment. Bull Seismol Soc Am 95(2):377– 389 Bommer JJ, Douglas J, Scherbaum F, Cotton F, Bungum H, Fäh D (2010) On the selection of ground-motion prediction equations for seismic hazard analysis. Seismol Res Lett 81(5):794–801 Bommer JJ, Akkar S, Kale Ö (2011a) A model for vertical-to-horizontal response spectral ratios for Europe and the Middle East. Bull Seismol Soc Am 101(4) (in press) Bommer JJ, Papaspiliou M, Price W (2011b) Earthquake response spectra for seismic design of nuclear power plants in the UK. Nucl Eng Des 241(3):968–977 Boore DM, Atkinson GM (2008) Ground-motion prediction equations for the average horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between 0.1 s and 10.0 s. Earthq Spectra 24(1):99–138 Boore DM, Bommer JJ (2005) Processing strong-motion accelerograms: needs, options and consequences. Soil Dyn Earthq Eng 25(2):93–115 Bozorgnia Y, Campbell KW (2004) The vertical-to-horizontal spectral ratio and tentative procedures for developing simplified V/H and vertical design spectra. J Earthq Eng 4(4):539–561 Brady AG, Trifunac MD, Hudson DE (1973) Analyses of strong motion earthquake accelerograms—response spectra. EERC Technical Report, Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena Campbell KW (2003) Prediction of strong ground motion using the hybrid empirical method and its use in the development of ground-motion (attenuation) relations in Eastern North America. Bull Seismol Soc Am 93(3):1012–1033

123

398


Campbell KW, Bozorgnia Y (2008) NGA ground motion model for the geometric mean horizontal component of PGA, PGV, PGD and 5%-damped linear elastic response spectra at periods ranging from 0.1 s to 10.0 s. Earthq Spectra 24(1):139–171 Cauzzi C, Faccioli E (2008) Broadband (0.05 to 20 s) prediction of displacement response based on worldwide digital records. J Seismol 12:453–475 Chiou BS-J, Youngs RR (2008) An NGA model for the average horizontal component of peak ground motion and response spectra. Earthq Spectra 24(1):173–215 Chiou B, Darragh R, Gregor N, Silva W (2008) NGA project strong-motion database. Earthq Spectra 24(1):23–44 Cotton F, Pousse G, Bonilla F, Scherbaum F (2008) On the discrepancy of recent European ground-motion observations and predictions from empirical models: Analysis of KiK-net accelerometric data and pointsources stochastic simulations. Bull Seismol Soc Am 98(5):2244–2261 Danciu L, Tselentis G-A (2007) Engineering ground-motion parameters attenuation relationships for Greece. Bull Seismol Soc Am 97(1B):162–183 Douglas J, Bungum H, Scherbaum F (2006) Ground-motion prediction equations for southern Spain and southern Norway obtained using the composite hybrid model perspective. J Earthq Eng 10(1):33–72 Douglas J, Boore DM (2011) High-frequency filtering of strong-motion records. Bull Earthquake Eng 9(2):395–409 Elnashai AS, Papazoglu AJ (1997) Procedure and spectra for analysis of RC structures subjected to vertical earthquake loads. J Earthq Eng 1(1):121–155 García D, Singh SK, Herráiz M, Ordaz M, Pacheco JF (2005) Inslab earthquakes of Central Mexico: peak ground-motion parameters and response spectra. Bull Seismol Soc Am 95(6):2272–2282 Gülerce Z, Abrahamson NA (2011) Site-specific spectra for vertical ground motion. Earthq Spectra (in press) Idriss IM (2007) Empirical models for estimating the average horizontal values of pseudo-absolute spectral accelerations generated by crustal earthquakes, vol 1: sites with Vs30 = 450 to 900 m/s. Interim report issues for USGS review, PEER Center, University of California at Berkeley, 19 January Idriss IM (2008) An NGA empirical model for estimating the horizontal spectral values generated by shallow crustal earthquakes. Earthq Spectra 24(1):217–242 Joyner WB, Boore DM (1993) Methods for regression analysis of strong-motion data. Bull Seismol Soc Am 83:469–487 Kalkan E, Gülkan P (2004) Site-dependent spectra derived from ground motion records in Turkey. Earthq Spectra 20(4):111–1138 Kanno T, Narita A, Morikawa N, Fujiwara H, Fukushima Y (2006) A new attenuation relation for strong ground motion in Japan based on recorded data. Bull Seismol Soc Am 96(3):879–897 Lin P-S, Lee C-T (2008) Ground-motion attenuation relationships for subduction-zone earthquakes in northeastern Taiwan. Bull Seismol Soc Am 98(1):220–240 Massa M, Morasca P, Moratto L, Marzorati S, Costa G, Spallarossa D (2008) Empirical ground-motion equations for northern Italy using weak- and strong-motion amplitudes, frequency content, and duration parameters. Bull Seismol Soc Am 98(3):1319–1342 McGuire RK (1977) Seismic design spectra and mapping procedures using hazard analysis based directly on oscillator response. Earthq Eng Struct Eng 5:211–234 McVerry G, Zhao J, Abrahamson N, Somerville P (2006) New Zealand acceleration response spectrum attenuation relations for crustal and subduction zone earthquakes. Bull New Zealand Soc Earthq Eng 39(1):1–58 Özbey C, Sari A, Manuel L, Erdik M, Fahjan Y (2004) An empirical attenuation relationship for northwestern Turkey ground motion using a random effects approach. Soil Dyn Earthq Eng 24:115–125 Pankow KL, Pechmann JC (2004) The SEA99 ground-motion predictive relations for extensional tectonic regimes: revisions and a new peak ground velocity relation. Bull Seismol Soc Am 94(1):341–348 Sabetta F, Pugliese A (1996) Estimation of response spectra and simulation of nonstationary earthquake ground motions. Bull Seismol Soc Am 86(2):337–352 Tavakoli B, Pezeshk S (2005) Empirical-stochastic ground-motion prediction for Eastern North America. Bull Seismol Soc Am 95(6):2283–2296 Toro GR, Abrahamson NA, Schneider JF (1997) Models of strong ground motions from earthquakes in Central and Eastern North America: best estimates and uncertainties. Seismol Res Lett 68(1):41–57 USNRC (2008) Interim staff guidance on seismic issues associated with high-frequency ground motions in design certification and combined license applications. ISG-01, US Nuclear Regulatory Commission, Washington, DC Youngs RR, Chiou S-J, Silva WJ, Humphreys JR (1997) Ground motion attenuation relationships for subduction zone earthquakes. Seismol Res Lett 68(1):58–73

123


399

Zhao JX, Zhang J, Asano A, Ohno Y, Oouchi T, Takahashi T, Ogawa H, Irikura K, Thio HK, Somerville PG, Fukushima Y, Fukushima Y (2006) Attenuation relations of strong ground motion in Japan using site classifications based on predominant period. Bull Seismol Soc Am 96(3):898–913

123